What is the ruling on a woman keeping her husband's family name, and must she revert to her father's family name? And does a child born out of wedlock carry the family name of his father or his mother, and what is the evidence for that?

It is not permissible for a person to ascribe himself to other than his father. The agreement with the disbelievers in adding the wife's lineage to her husband and removing her father's lineage is unlawful, false, and an insult to the woman. Repentance is obligatory, and the right must be restored to its proper place by attributing oneself to one's legitimate father. As for the child of fornication, he is attributed to his mother and not added to the lineage of the fornicator.